2.
␣ϭ
1
Z
e2
m
͑␻0Ϫ␥͒2
Ϫ␻2
Ϫi␥␻
͑͑␻0Ϫ␥͒2
Ϫ␻2
͒2
ϩ␥2
␻2
ϩ
1
Z
e2
m
͑␻0ϩ␥͒2
Ϫ␻2
ϩi␥␻
͑͑␻0ϩ␥͒2
Ϫ␻2
͒2
ϩ␥2
␻2
Ϸ
1
Z
e2
m
␻0
2
Ϫ2␥␻0Ϫ␻2
Ϫi␥␻
͑␻0
2
Ϫ2␥␻0Ϫ␻2
͒2
ϩ␥2
␻2
ϩ
1
Z
e2
m
␻0
2
ϩ2␥␻0Ϫ␻2
ϩi␥␻
͑␻0
2
ϩ2␥␻0Ϫ␻2
͒2
ϩ␥2
␻2 , ͑8͒
where the approximation is obtained by the neglect of terms
in ␥2
compared to those in ␥␻0 .
We now consider the issue of attenuation of a pulse of
frequency ␻. Since kϭ␻n/cϷ␻(1ϩ2␲N␣)/c, the spatial
dependence eikz
of a pulse propagating in the z direction
includes attenuation if the imaginary part of the index n is
nonzero. However, the population inversion described by
␥2ϭϪ␥1 leads to Im͓␣(␻0)͔ϭ0. Hence, there is no attenua-
tion of a probe pulse at frequency ␻0 .
In the present model, the pulse is attenuated at frequencies
less than ␻0 , but grows ͑lases͒ at frequencies greater than
␻0 . In the experiment of Hau et al.,1
lasing did not occur
because line 2 actually corresponded to a transition between
the upper level of line 1 and a third, excited level. ͑In a
sense, the quantum mechanical level structure with one high
and two low energy levels is the inverse of that assumed in
the classical model here, i.e., one low and two high levels.͒
Therefore, pumping at frequency ␻2 did not produce an in-
verted population that could lead to lasing; but it did lead to
an effective sign reversal of the damping constant ␥2 for a
narrow range of frequencies near ␻0 .
To obtain the group velocity at frequency ␻0 , we need the
derivative
d Re͑n͒
d␻
ͯ␻0
ϭ2␲N
d Re͑␣͒
d␻
ͯ␻0
ϭ
24␲Ne2
25Zm␥2
␻0
. ͑9͒
Since ␣(␻0)ϭ0, we have n(␻0)ϭ1, and the phase velocity
at ␻0 is exactly c. The group velocity ͑2͒ is
vgϭ
c
1ϩ
24␲Ne2
25Zm␥2
Ϸ
25Z␥2
24␲N
e2
mc2 c2
cϷ
Z␥2
␲Nr0c
, ͑10͒
where r0ϭe2
/mc2
Ϸ3ϫ10Ϫ13
cm is the classical electron ra-
dius. The group velocity is lower in a denser medium.
In the experiment of Hau et al., the medium was sodium
vapor (Zϭ11), cooled to less than 1 ␮K to increase the
density. An additional increase in density by a factor of 5
was obtained when the vapor formed a Bose condensate.
Plugging in the experimental parameters, Nϭ5ϫ1012
/cm3
and ␥ϭ5ϫ106
/s, we ﬁnd
vgϷ
11•͑5ϫ106
͒2
3•5ϫ1012
•3ϫ10Ϫ13
•3ϫ1010 Ϸ2000 cm/s, ͑11͒
compared to the measured value of 1700 cm/s.
1
L. V. Hau et al., ‘‘Light speed reduction to 17 metres per second in an
ultracold atomic gas,’’ Nature ͑London͒ 397, 594–598 ͑1999͒.
MAKE AN IMPORTANT DISCOVERY!
Make an important discovery, and you are a successful scientist in the true, elitist sense in a
profession where elitism is practiced without shame. You go into the textbooks. Nothing can take
that away; you may rest on your laurels the rest of your life. But of course you won’t. Almost no
one driven enough to make an important discovery ever rests. And any discovery at all is thrilling.
There is no feeling more pleasant, no drug more addictive, than setting foot on virgin soil.
Edward O. Wilson, ‘‘Scientists, Scholars, Knaves and Fools,’’ Am. Scientist 86 ͑1͒, 6–7 ͑1998͒.
294 294Am. J. Phys., Vol. 68, No. 3, March 2000 New Problems